I assume that you have already tried to talk to this neighbor about the issue and have not been able to get cooperation. If not, I'd start by trying to resolve this with the dog owner. If he refuses, send him a complaint letter by certified mail and send a copy to your landlord. Make sure you do it as "return receipt" so you will have evidence that it was received. Also, take photos of the unpleasant droppings. This may prove valuable as evidence later on.
You live on the bottom and the dog is on the top floor. How about the neighbor above you? Have you talked to them? They may have similar complaints and if both of you complained to the landlord it may have more impact. If it is not taken care of to your satisfaction, personally I would get on the phone with your local Health Department and the SPCA and put in a complaint. They will come and investigate. This is a serious potential health issue for everyone (human and animal) and you should absolutely not have to put up with this...and neither should your dog! If they won't help, try the SPCA in your area. Not providing adequate "facilities" for a dog to do his business would be considered neglect and the SPCA should be able to help you.
In Massachusetts where I live Tenant Rights are on the web and also available in booklet form. They may also be available for your state. If you can't find them by a web search, call your city hall for help. You may be able to legally withhold rent until the situation is resolved.